Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: 𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvdmsscn.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
Ref | Expression |
---|---|
dvdmsscn | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 16705 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 | |
2 | dvdmsscn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | 1, 2 | sseldi 3965 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
4 | elpwi 4548 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | dvdmsscn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
7 | recnprss 24502 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 5, 8 | sstrd 3977 | 1 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 {cpr 4569 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 ↾t crest 16694 TopOpenctopn 16695 ℂfldccnfld 20545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-rest 16696 |
This theorem is referenced by: dvxpaek 42245 etransclem17 42556 etransclem18 42557 etransclem20 42559 etransclem21 42560 etransclem22 42561 etransclem29 42568 etransclem31 42570 etransclem34 42573 etransclem43 42582 etransclem46 42585 |
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